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Metamath Proof Explorer

Theorem cdleme8

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. C represents s_1. In their notation, we prove p \/ s_1 = p \/ s. (Contributed by NM, 9-Jun-2012)

Ref Expression
Hypotheses cdleme8.l 
|- .<_ = ( le ` K )
cdleme8.j 
|- .\/ = ( join ` K )
cdleme8.m 
|- ./\ = ( meet ` K )
cdleme8.a 
|- A = ( Atoms ` K )
cdleme8.h 
|- H = ( LHyp ` K )
cdleme8.4 
|- C = ( ( P .\/ S ) ./\ W )
Assertion cdleme8 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ C ) = ( P .\/ S ) )

Proof

Step Hyp Ref Expression
1 cdleme8.l 
 |-  .<_ = ( le ` K )
2 cdleme8.j 
 |-  .\/ = ( join ` K )
3 cdleme8.m 
 |-  ./\ = ( meet ` K )
4 cdleme8.a 
 |-  A = ( Atoms ` K )
5 cdleme8.h 
 |-  H = ( LHyp ` K )
6 cdleme8.4 
 |-  C = ( ( P .\/ S ) ./\ W )
7 6 oveq2i 
 |-  ( P .\/ C ) = ( P .\/ ( ( P .\/ S ) ./\ W ) )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. HL )
9 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P e. A )
10 8 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. Lat )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 9 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P e. ( Base ` K ) )
14 11 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
15 14 3ad2ant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> S e. ( Base ` K ) )
16 11 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ S ) e. ( Base ` K ) )
17 10 13 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
18 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> W e. H )
19 11 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> W e. ( Base ` K ) )
21 11 1 2 latlej1 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> P .<_ ( P .\/ S ) )
22 10 13 15 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P .<_ ( P .\/ S ) )
23 11 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) )
24 8 9 17 20 22 23 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) )
25 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
26 1 2 25 4 5 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = ( 1. ` K ) )
27 26 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ W ) = ( 1. ` K ) )
28 27 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( ( P .\/ S ) ./\ ( P .\/ W ) ) = ( ( P .\/ S ) ./\ ( 1. ` K ) ) )
29 hlol 
 |-  ( K e. HL -> K e. OL )
30 8 29 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. OL )
31 11 3 25 olm11 
 |-  ( ( K e. OL /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) )
32 30 17 31 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) )
33 24 28 32 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( P .\/ S ) )
34 7 33 eqtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ C ) = ( P .\/ S ) )